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Here each $p_j(n)$ is a polynomial. The main result is the following: for the characteristic polynomial $c(t)=t^k+c_1t^{k-1}+\\cdots+c_k$, if $s_j$ denotes the multiplicity of $r_j$ as a root of $c(t)$ ($s_j=0$ when $r_j$ is not a root), then there exists a particular solution of the form $q_n=\\sum_{j=1}^J b_j(n)n^{s_j}r_j^n$, where each $b_j"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.04700","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-06T06:04:03Z","cross_cats_sorted":[],"title_canon_sha256":"6511bfb37d67d6f590152809bea6d14b0bcf12b80c58dcf64e510d7132cf9282","abstract_canon_sha256":"e45e1c9124410221cb49477bbe63b987754981f2ef2cc9caa31beca2a64cbe92"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:19:58.745968Z","signature_b64":"9QIllS2yNlC7nuZ2dIObgkDFF3WgDycVGkCVztWvLWYgdpbyycw4qMih5Z7xVPFVeSzgKKrVOnM202w2oCm1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8181b5ab104d69656121759132b5b630c6f8c7ffa96744e984cb33159ab18bfd","last_reissued_at":"2026-07-07T02:19:58.745192Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:19:58.745192Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Determining Particular Solutions for Exponential-Polynomial Forcing Terms in Linear Nonhomogeneous Recurrence Relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heesung Shin","submitted_at":"2026-07-06T06:04:03Z","abstract_excerpt":"This paper develops a systematic method for determining particular solutions of the $k$th-order linear nonhomogeneous recurrence relation $$a_n + c_1 a_{n-1} + \\cdots + c_k a_{n-k} = \\sum_{j=1}^J p_j(n){r_j}^n$$ with $n \\geq k$, $c_k \\neq 0$, $r_j \\neq 0$. Here each $p_j(n)$ is a polynomial. The main result is the following: for the characteristic polynomial $c(t)=t^k+c_1t^{k-1}+\\cdots+c_k$, if $s_j$ denotes the multiplicity of $r_j$ as a root of $c(t)$ ($s_j=0$ when $r_j$ is not a root), then there exists a particular solution of the form $q_n=\\sum_{j=1}^J b_j(n)n^{s_j}r_j^n$, where each $b_j"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04700","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.04700/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.04700","created_at":"2026-07-07T02:19:58.745325+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.04700v1","created_at":"2026-07-07T02:19:58.745325+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.04700","created_at":"2026-07-07T02:19:58.745325+00:00"},{"alias_kind":"pith_short_12","alias_value":"QGA3LKYQJVUW","created_at":"2026-07-07T02:19:58.745325+00:00"},{"alias_kind":"pith_short_16","alias_value":"QGA3LKYQJVUWKYJB","created_at":"2026-07-07T02:19:58.745325+00:00"},{"alias_kind":"pith_short_8","alias_value":"QGA3LKYQ","created_at":"2026-07-07T02:19:58.745325+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD","json":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD.json","graph_json":"https://pith.science/api/pith-number/QGA3LKYQJVUWKYJBOWITFNNWGD/graph.json","events_json":"https://pith.science/api/pith-number/QGA3LKYQJVUWKYJBOWITFNNWGD/events.json","paper":"https://pith.science/paper/QGA3LKYQ"},"agent_actions":{"view_html":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD","download_json":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD.json","view_paper":"https://pith.science/paper/QGA3LKYQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.04700&json=true","fetch_graph":"https://pith.science/api/pith-number/QGA3LKYQJVUWKYJBOWITFNNWGD/graph.json","fetch_events":"https://pith.science/api/pith-number/QGA3LKYQJVUWKYJBOWITFNNWGD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD/action/storage_attestation","attest_author":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD/action/author_attestation","sign_citation":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD/action/citation_signature","submit_replication":"https://pith.science/pith/QGA3LKYQJVUWKYJBOWITFNNWGD/action/replication_record"}},"created_at":"2026-07-07T02:19:58.745325+00:00","updated_at":"2026-07-07T02:19:58.745325+00:00"}